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Wave Packets. Recall that for a wave packet x k~1 to localize a wave to some region x we need a spread of wavenumbers k de Broglie hypothesis =h/p or p=h/ but k=2/ and p=(h/2)(2/) = ħ k h bar p= ħ k and x p~ ħ x p ħ uncertainty principle.
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Wave Packets • Recall that for a wave packet x k~1 • to localize a wave to some region x we need a spread of wavenumbers k • de Broglie hypothesis =h/p or p=h/ • but k=2/ and p=(h/2)(2/) = ħk • h bar • p= ħkand x p~ ħ • x p ħ uncertainty principle
Heisenberg’s Uncertainty Principle • States that measured values cannot be assigned to the position r and momentum p of a particle simultaneously with unlimited precision • introduce “h-bar” ħ = h/2 • x px ħ • y py ħ • z pz ħ
Energy-Time Uncertainty • Recall for photons E = hf • uncertainty in energy => uncertainty in f • f= E/h = 2f .t 1 • introduce “h-bar” ħ = h/2 • E . t ħ • need infinite time to make an accurate measurement of energy • eg. Ground state of atom has well defined energy • excited state does not
Uncertainty Principle • If px= py= pz =0 (free particle) • then x = y = z = • an electron has kinetic energy K = 12.0 eV • K=(1/2)mv2 = 12.0 eV => v = 2.05 x 106 m/s • suppose the electron is moving // x and we can measure its speed with .50% precision (one part in 200) • what is the uncertainty in its position?
px=mv =(.911x 10-31 kg)(2.05 x 106 m/s) = 1.87 x 10-24 kg.m/s • px = .005 (1.87 x 10-24 kg.m/s) = 9.35 x10-27 kg.m/s • x~ ħ/px =(6.63 x 10-34 J.s)/(2)(9.35 x10-27 kg.m/s) = 11.0 nm ~ 100 atomic diameters
Problem • Imagine a universe in which Planck’s constant h = .60 J.s • quantum baseball • consider a .50 kg baseball that is moving at 20 m/s along an axis • if the uncertainty in its speed is 0.5 m/s, what is the uncertainty in its position? • p = (.5 kg)(0.5 m/s) = .25 kg.m/s • x ħ/(p) = (.60/2)/.25 = .38 m
Imaginary part Real part Wave function • What kind of function is (x,y,z,t)? • Travelling wave? Standing wave? • Let’s learn something about complex numbers • sqrt(-1) is a complex number i2 = -1 • a complex number has the general form a + i b • ‘a’ is the real part and ‘b’ is the imaginary part • a complex function has the form f1(x) + i f2(x)
Imaginary axis b a Real axis z = a + i b z* = a - i b Complex conjugate Modulus |a + i b| = sqrt (a2 + b2) zz* = (a + ib)(a -ib) = a2 - i2 b2 = a2 + b2 = |z|2
sin cos Complex function • ex = 1 +x + x2/2 + x3/6 + … + xn/n! +… • e i = 1 +i +(i)2/2 +(i )3/6 +(i )4/24 + … • = 1 + i - 2/2 -i 3/6 +()4/24 + … • = (1 - 2/2 + 4/24 - …) +i( - 3/6 +…) • = cos + i sin • e-i = cos - i sin e i= cos + i sin
Complex Waves • (x,t) = f(x) g(t) a standing wave • string: y(x,t)=2ymsin(kx) cos(t) • matter wave: (x,t) = f(x) e-it • = f(x)[ cos t - i sin t ]
P.E. Rest energy K.E. Schrödinger’s Quantum Mechanics • Probability waves are described by a wave function (x,y,z,t) • Schrödinger adopted de Broglie’s equations • =h/p wavelength of a particle • f = E/h frequency of a particle • non-relativistic energy is • E = mc2 + p2/2m + V
Quantum Mechanics • Choose mc2 as the reference energy • E = p2/2m + V particle • hf = h2/2m2 + V wave • relates f to • what is the equation for the wave function (x,t) ?
Schrödinger Equation • Schrödinger Equation 1926 • (x,t) is a solution of this equation • the wave equation for matter waves
Free Particle • Consider the case of V=0 • hence E=(1/2)mv2 = p2/2m
Free Particle • But p=h/ and k=2/ Can we solve this equation?